Research ArticlePHYSICS

First-order synchronization transition in a large population of strongly coupled relaxation oscillators

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Science Advances  23 Sep 2020:
Vol. 6, no. 39, eabb2637
DOI: 10.1126/sciadv.abb2637
  • Fig. 1 Coupled photochemical oscillators.

    (A) Experimental setup. A thermostatted open reactor, hosting 2600 chemical oscillators, is spectrophotometrically monitored in fluorescence light (λ > 550 nm) with a CMOS camera. Recorded light intensities fi determine the photochemical feedback Ii, which is applied with a spatial light modulator. (B) Camera image of the fluorescing oscillator reservoir. The connectivity between oscillators is overlaid in blue. (C) Hysteresis loop of the Kuramoto order parameter R in the case of N = 1000 all-to-all coupled oscillators with unimodally distributed natural frequencies (SD, σω = 0.012 rad/s).

  • Fig. 2 Experimental observation of hysteresis in the case of N = 1000 all-to-all coupled oscillators with normally distributed natural frequencies.

    (A) Time protocol for the coupling strength K. (B) Time evolution of the instantaneous frequencies (ω¯i). The color of each line corresponds to the natural frequency of the nodes (ωi), respectively. The oscillators synchronize in-phase at K = 0.4 but transition back to incoherence at K = 0.1 < K (see also Fig. 1C). (C to E) Fluorescence value (fi) plots for clustering, synchronized, and incoherent states as observed during the cycle. The oscillators are indexed (i) in order of increasing natural frequencies. Synchronization from incoherent initial conditions (E) proceeds with the formation of antiphase clusters (C), which delay the onset of synchronization (D) leading to hysteretic behavior. The antiphase clustering state is characterized by a higher degree of frequency coherence for low-frequency oscillators.

  • Fig. 3 Experimental observation of hysteresis for a bimodal frequency distribution.

    (A) Time protocol for coupling strength K. (B) Time evolution of the instantaneous frequencies (ω¯i) of N = 200 oscillators. The color of each line encodes the natural frequency of the nodes (ωi), respectively. The oscillators synchronize in-phase at K = 0.72 upon increasing K but transition back to incoherence at K = 0.56 < K upon decreasing K. (C to E) Individual fluorescence values for clustering (C), synchronized (D), and incoherent states (E) during the experiment. The oscillators are indexed (i) in order of increasing natural frequencies. Synchronization from incoherent initial conditions proceeds with the formation of approximately antiphase clusters, which delay the onset of synchronization leading to hysteresis.

  • Fig. 4 First-order synchronization transitions in all-to-all coupled oscillator populations.

    Hysteresis curves for the order parameter (R) with increasing (blue) or decreasing (orange) coupling strength in chemical experiments (A and B) and numerical simulations (C and D). We consider N = 200 globally coupled oscillators with unimodal (A and C) or bimodal (B and D) natural frequency distributions. Because of finite-size effects, RO(1/N) in the unsynchronized phase (low K).

  • Fig. 5 Relaxation character determines the order of the synchronization transition.

    (A and B) PRCs of the FHN model for time scale separation parameter ϵ = 0.3 (A) with perturbation strengths of 0.4, 0.6, and 0.8 (light-orange, purple, and light blue, respectively) and ϵ = 10 (B) with perturbation strengths of 0.5, 1.0, and 1.5 (light orange, purple, and light blue, respectively). The phase response curves are normalized by the respective perturbation strengths in (A). (C) Phase response curves determined from chemical experiments (dots) together with functions fitted with model (3) for perturbing light intensities of 0.01 (light orange), 0.06 (purple), and 0.25 mW cm−2 (light blue). (D and E) Order parameter curves in the case of N = 200 globally coupled FHN oscillators with unimodally distributed natural frequencies for ϵ = 0.3 (D) and ϵ = 10 (E). (F) Schematic representation of the mechanism of first-order synchronization via antiphase cluster formation: time evolution of two oscillators from different subpopulations (v1,v2) and their excitable intervals (hatched regions) together with the mean amplitude (v¯) in antiphase (★) and in-phase synchronized (◆) states.

Supplementary Materials

  • Supplementary Materials

    First-order synchronization transition in a large population of strongly coupled relaxation oscillators

    Dumitru Călugăru, Jan Frederik Totz, Erik A. Martens, Harald Engel

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